The present invention is directed, in general, to modeling and simulation systems and, more specifically, to an apparatus and method for reduced-order modeling of time-varying systems and a computer-readable storage medium containing the same.
An important task in communication system design is hierarchical verification of functionality at different levels, starting from individual circuits up to block representations of full systems. A key step in this process is the creation of small macromodels that abstract, to a given accuracy, the behavior of much bigger subsystems. The macromodels are then clustered into bigger blocks and the procedure may be repeated at a higher level. For systems with xe2x80x9cnonlinearxe2x80x9d blocks like mixers and switched-capacitor filters, this is typically achieved by using results from detailed nonlinear simulations to construct macromodels manually (based on an understanding of the operation of the blocks in question). This process has disadvantages. Nonlinear simulation does not provide parameters of interest (such as poles and zeros) directly; to obtain them by inspection, frequency-response plots with many points are often computed. This can be very time-consuming for large subsystems, since nonlinear blocks require a steady-state solution at each point. Also, the macromodeling step, critical for reliable verification, is heuristic, time-consuming and highly reliant on detailed internal knowledge of the system under consideration.
Reduced-order modeling is well established for circuit applications. For example, L. T. Pillage and R. A. Rohrer, Asymptotic Waveform Evaluation for Timing Analysis, IEEE Trans. CAD, 9:352-366, April 1990; X. Huang, V. Raghavan, and R. A. Rohrer, AWEsim: A Program for the Efficient Analysis of Linear(ized) Circuits, Proc. ICCAD, pages 534-537, November 1990; and E. Chiprout and M. S. Nakhla. Asymptotic Waveform Evaluation Kluwer, Norwell, Mass., 1994, address one technique (AWE). P. Feldman and R. Freund, Efficient Linear Circuit Analysis by Padxc3xa9 Approximation via the Lanczos Process, IEEE Tans. Cad, 14(5):639-649, May 1995; and P. Feldmann and R. Freund, Reduced-Order Modeling of Large Linear Subcircuits via a Block Lanczos Algorithm, Proc. IEEE DAC, pages 474-479, 1995, address another technique (PVL). Finally, A. Odabasioglu, M. Celik, and L. T. Pileggi, PRIMA: Passive Reduced-Order Interconnect Macromodeling Algorithm, Proc. ICCAD, pages 58-65, 1997, address still a third technique (PRIMA). (These articles are incorporated herein by reference.) However, these existing methods are applicable only to linear time-invariant (LTI) systems. They are inadequate for communication blocks with properties such as frequency translation, that cannot be represented by LTI models.
Linear time-varying (LTV) descriptions of a system, on the other hand, can capture frequency translation and mixing/switching behavior. LTV transfer functions are often computed in the context of radio frequency (RF) simulation (e.g., plotting frequency-responses or calculating cyclostationary noise). See, for example, R. Telichevesky, K. Kundert, and J. White, Efficient AC and Noise Analysis of Two-Tone RF Circuits, Proc. IEEE DAC, pages 292-297, 1996, incorporated herein by reference. However, a formulation suitable for reducing the order of time-varying transfer functions is not, at this point in time, available. Accordingly, what is needed in the art is a way of generating a time-varying transfer function that is amenable to order reduction.
To address the above-discussed deficiencies of the prior art, the present invention provides an apparatus for, and method of, modeling a time-varying system and a computer-readable storage medium containing the apparatus or the method. In one embodiment, the apparatus includes: (1) a transfer function generator that develops, for the system, a linear, time-varying transfer function of a particular order and having separate scales corresponding to time variations in the system and an input thereto and (2) an approximator, coupled to the transfer function generator, that approximates the transfer function to yield a model of an order lower than the particular order.
The basic difficulty heretofore in generalizing LTI model-reduction techniques to the LTV case has been the interference of system time variations with input time variations. The present invention recognizes that time scales should be kept separate, using recently-developed concepts of multiple time variables and multirate partial differential equations (MPDE), resulting in forms for the LTV transfer function that are suitable for model reduction.
The present invention therefore introduces the broad concept of developing a transfer function for a time-varying system in which the system-based and input-based time variations present in the system are kept separate. This allows an approximation to be applied to the transfer function while preserving the independence of the time variations. Thus, approximations which were heretofore useful only in conjunction with time-invariant systems, can now be employed to model time-variant systems, including those that contain nonlinearities.
In one embodiment of the present invention, the approximator is a Padxc3xa9 approximator, the model containing a Padxc3xa9 approximation of the transfer function. Those skilled in the pertinent art are familiar with Padxc3xa9 approximations and their use in approximating time-invariant functions. The present invention employs such approximations to advantage in modeling time-varying systems. However, the present invention is not limited to Padxc3xa9-type approximations.
In one embodiment of the present invention, the order of the model is selectable. To Thus, a user can select the extent to which the approximation simplifies the transfer function. Of course, while approximations of greater order may be more accurate to the actual transfer function, they are generally more complex and consume more computational time to calculate.
In one embodiment of the present invention, the approximator explicitly calculates moments of the transfer function and matches the moments to yield the model. In an alternative embodiment, the approximator replaces the transfer function directly with the model. Moments of the transfer function are therefore matched implicitly. In a more particular case of the latter embodiment, the approximator replaces the transfer function by employing a Krylov-subspace process. Several different ways of approximating the transfer function will be set forth and explained in detail in the Detailed Description that follows.
The foregoing has outlined, rather broadly, preferred and alternative features of the present invention so that those skilled in the pertinent art may better understand the detailed description of the invention that follows. Additional features of the invention will be described hereinafter that form the subject of the claims of the invention. Those skilled in the pertinent art should appreciate that they can readily use the disclosed conception and one or more specific embodiments as a basis for designing or modifying other structures for carrying out the same purposes of the present invention. Those skilled in the pertinent art should also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form or the claims.